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%I #18 Mar 04 2018 17:47:17
%S 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,12,1,2,1,1,1,1,12,1,2,1,1,
%T 1,1,1,48,1,2,1,2,1,1,1,1,144,1,2,1,2,1,1,1,1,1,1440,3,8,1,12,1,2,1,1,
%U 1,1,1440,3,8,1,12,1,2,1,1,1,1,1,17280,3,80
%N T(n, k) = Product{1<=j<=n, gcd(j,k)=1 | j} / lcm{1<=j<=n, gcd(j,k)=1 | j} for n >= 0, k >= 1, square array read by antidiagonals.
%C T(n,k) = Product(R(n,k))/lcm(R(n,k)) where R(n,k) is the set of all integers up to n that are relatively prime to k.
%C T(n,k) = A216919(n,k)/A216917(n,k).
%F For n > 0:
%F A(n,1) = A025527(n);
%F A(4,n) = A000034(n);
%F A(n,n) = A128247(n).
%e k |n=0 1 2 3 4 5 6 7 8 9 10
%e ---+------------------------------------
%e 1 | 1 1 1 1 2 2 12 12 48 144 1440
%e 2 | 1 1 1 1 1 1 1 1 1 3 3
%e 3 | 1 1 1 1 2 2 2 2 8 8 80
%e 4 | 1 1 1 1 1 1 1 1 1 3 3
%e 5 | 1 1 1 1 2 2 12 12 48 144 144
%e 6 | 1 1 1 1 1 1 1 1 1 1 1
%e 7 | 1 1 1 1 2 2 12 12 48 144 1440
%e 8 | 1 1 1 1 1 1 1 1 1 3 3
%e 9 | 1 1 1 1 2 2 2 2 8 8 80
%e 10 | 1 1 1 1 1 1 1 1 1 3 3
%e 11 | 1 1 1 1 2 2 12 12 48 144 1440
%e 12 | 1 1 1 1 1 1 1 1 1 1 1
%e 13 | 1 1 1 1 2 2 12 12 48 144 1440
%o (Sage)
%o def A216915(n, k):
%o def R(n, k): return [j for j in (1..n) if gcd(j, k) == 1]
%o return mul(R(n,k))/lcm(R(n, k))
%o for k in (1..13): [A216915(n, k) for n in (0..10)]
%K nonn,tabl
%O 1,11
%A _Peter Luschny_, Oct 02 2012
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